Visualization of the Model:
Multiple Linear Regression
Introduction
Recall the general linear model, y = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k
\beta_0 is the y-intercept, or the average outcome (y) when all x_i = 0.
\beta_i is the slope for predictor i and describes the relationship between the predictor and the outcome, after adjusting (or accounting) for the other predictors in the model.
In the last lecture, we used a linear model to explore the relationships between clubhouse happiness and laughs, grumbles, and time spent with friends.
Lecture Example Set Up
On a busy day at the clubhouse, Mickey Mouse wants to understand what drives “happiness” at the end of the day. For each day, he records (in the clubhouse dataset):
- Time with friends (in hours; time_with_friends): how many hours Mickey spends hanging out with his friends.
- Goofy Laughs (a count; goofy_laughs) – how many big goofy laughs happen that day.
- Donald Grumbles (a count; donald_grumbles): how many times Donald gets frustrated and grumbles.
- Clubhouse Happiness (a score; clubhouse_happiness): an overall happiness score at the end of the day.
Example 1: Simple Linear Regression
- For our first example, let’s look at clubhouse happiness as a function of time spent with friends.
m1 <- glm(clubhouse_happiness ~ time_with_friends,
family = "gaussian",
data = clubhouse)
m1 %>% tidy()\hat{\text{happiness}} = 57.55 + 3.49 \text{ time}
Model Visualization: Simple Linear Regression
We can visualize this simple linear regression model with a scatterplot and regression line.
To create the regression line, we need to create predicted values from our model.
- Now we can plot the data and the regression line.
Plotting using library(ggplot2)
- The
ggplot()function initializes a ggplot object.
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
- We must define the aesthetics (i.e., the x and y variables) inside
ggplot().
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
We must add
geom_TYPE()layers to actually see anything on the plot.- We layer multiple
geoms to the plot using+operator.
- We layer multiple
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
- We continue to layer our plot with additional
geom_TYPE()s,
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
Ooops! That
geom_line()didn’t work as expected.- In the
ggplot(), we set y to be the actual happiness values (clubhouse_happiness). - We now need to overwrite the y variable to be the the predicted values from our model (predicted_happiness).
- In the
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
Now, we can work on “prettying” up our plot.
- My first step is to change the theme using
theme_NAME().
- My first step is to change the theme using
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
Now, we can work on “prettying” up our plot.
- Then, I want to clean up the axis titles.
- In our example,
Plotting using library(ggplot2)
Plotting using library(ggplot2)
Now, we can work on “prettying” up our plot.
- We could also add a graph title.
- In our example,
clubhouse %>% ggplot(aes(x = time_with_friends,
y = clubhouse_happiness)) +
geom_point() +
geom_line(aes(y = predicted_happiness_k1)) +
labs(x = "Time Spent with Friends (minutes)",
y = "Clubhouse Happiness",
title = "Predicted relationship between happiness and time spent with friends") +
theme_bw()Plotting using library(ggplot2)
Plotting using library(ggplot2)
Now, we can work on “prettying” up our plot.
- Outside of
aes(), we can specify colors, line types, point shapes, etc.
- Outside of
- In our example,
Plotting using library(ggplot2)
Example 2: Multiple Regression (k = 2)
- For our second example, let’s look at clubhouse happiness (clubhouse_happiness) as a function of time spent with friends (time_with_friends) and big, goofy laughs (goofy_laughs).
m2 <- glm(clubhouse_happiness ~ time_with_friends + goofy_laughs,
family = "gaussian",
data = clubhouse)
m2 %>% tidy()\hat{\text{happiness}} = 39.25 + 3.06 \text{ time} + 0.66 \text{ laughs}
Model Visualization: Multiple Regression
Now that there’s an additional predictor, we can’t easily visualize the model with a simple 2D scatterplot.
- In theory, we could create a 3D scatterplot with a regression plane, but those are hard to read and interpret.
Instead, we will visualize the relationship between y (clubhouse happiness) and x_1 (one predictor) while holding x_2 (the other predictor) constant.
In our example,
We will visualize the relationship between clubhouse happiness and time spent with friends.
Time spent with friends will be on the x-axis and allowed to vary.
We will hold goofy laughs constant at some value.
- With continuous predictors, I typically plug in the
median()when drafting initial graphs for collaborators.
- With continuous predictors, I typically plug in the
Model Visualization: Multiple Regression
Plotting using library(ggplot2)
- Then, constructing our graph,
Plotting using library(ggplot2)
Example 3: Multiple Regression (k = 3)
For our third example, let’s return to our full model.
We looked at clubhouse happiness (clubhouse_happiness) as a function of time spent with friends (time_with_friends), big, goofy laughs (goofy_laughs), and how much Donald grumbles (donald_grumbles).
m3 <- glm(clubhouse_happiness ~ time_with_friends + goofy_laughs + donald_grumbles,
family = "gaussian",
data = clubhouse)
m3 %>% tidy()\hat{\text{happiness}} = 47.58 + 3.58 \text{ time} + 0.66 \text{ laughs} - 1.06 \text{ grumbles}
Model Visualization: Multiple Regression
In this example, we have k=3 predictors.
- We can’t easily visualize the model with a simple 2D scatterplot or even a 3D scatterplot.
Instead, we will visualize the relationship between y (clubhouse happiness) and x_1 (one predictor) while holding all other x_i (the other predictors) constant.
- One x_i will vary on the x-axis.
- We will plug in plausible values for the other predictors.
In our example,
We will visualize the relationship between clubhouse happiness and time spent with friends.
Time spent with friends will be on the x-axis and allowed to vary.
We will hold goofy laughs constant at some value.
We will also hold Donald grumbles constant at some value.
Model Visualization: Multiple Regression
Plotting using library(ggplot2)
- Then, constructing our graph,
Plotting using library(ggplot2)
Let’s Explore…
- Hm… what if we put the three lines on top of one another? How different are the adjusted slopes?
Example 4: Multiple Regression (k = 3)
For our fourth example, let’s again return to our full model.
We looked at clubhouse happiness (clubhouse_happiness) as a function of time spent with friends (time_with_friends), big, goofy laughs (goofy_laughs), and how much Donald grumbles (donald_grumbles).
m4 <- glm(clubhouse_happiness ~ time_with_friends + goofy_laughs + donald_grumbles,
family = "gaussian",
data = clubhouse)
m4 %>% tidy()\hat{\text{happiness}} = 47.58 + 3.58 \text{ time} + 0.66 \text{ laughs} - 1.06 \text{ grumbles}
Model Visualization: Multiple Regression
Let’s now consider the relationship between clubhouse happiness and Donald’s grumbles.
- Donald grumbles will be on the x-axis and allowed to vary.
- We will hold goofy laughs constant at some value.
- We will also hold time spent with friends constant at some value.
Model Visualization: Multiple Regression
Plotting using library(ggplot2)
- Then, constructing our graph,
Plotting using library(ggplot2)
- Then, constructing our graph,
Wrap Up
In this lecture, we explored how to visualize simple and multiple linear regression models using the
ggplot2library.For simple linear regression, we visualized the relationship between the outcome and predictor using a scatterplot and regression line.
For multiple linear regression, we visualized the relationship between the outcome and one predictor while holding the other predictors constant.
Every week, we will review model visualization.
- The general ideas won’t change, but things will get tricky when we add categorical predictors and leave the normal distribution.
Next lecture: Model Assumptions and Diagnostics